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Theorem frgrancvvdeqlem3 30793
Description: Lemma 3 for frgrancvvdeq 30803. In a friendship graph, for each neighbor of a vertex there is exacly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx  |-  D  =  ( <. V ,  E >. Neighbors  X )
frgrancvvdeq.ny  |-  N  =  ( <. V ,  E >. Neighbors  Y )
frgrancvvdeq.x  |-  ( ph  ->  X  e.  V )
frgrancvvdeq.y  |-  ( ph  ->  Y  e.  V )
frgrancvvdeq.ne  |-  ( ph  ->  X  =/=  Y )
frgrancvvdeq.xy  |-  ( ph  ->  Y  e/  D )
frgrancvvdeq.f  |-  ( ph  ->  V FriendGrph  E )
frgrancvvdeq.a  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
Assertion
Ref Expression
frgrancvvdeqlem3  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  N  {
x ,  y }  e.  ran  E )
Distinct variable groups:    y, D    x, y, V    x, E, y    y, Y    ph, y
Allowed substitution hints:    ph( x)    A( x, y)    D( x)    N( x, y)    X( x, y)    Y( x)

Proof of Theorem frgrancvvdeqlem3
StepHypRef Expression
1 frgrancvvdeq.f . . . 4  |-  ( ph  ->  V FriendGrph  E )
21adantr 465 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  V FriendGrph  E )
3 frgrancvvdeq.nx . . . . . . 7  |-  D  =  ( <. V ,  E >. Neighbors  X )
43eleq2i 2532 . . . . . 6  |-  ( x  e.  D  <->  x  e.  ( <. V ,  E >. Neighbors  X ) )
5 frisusgra 30752 . . . . . . 7  |-  ( V FriendGrph  E  ->  V USGrph  E )
6 nbgraisvtx 23514 . . . . . . 7  |-  ( V USGrph  E  ->  ( x  e.  ( <. V ,  E >. Neighbors  X )  ->  x  e.  V ) )
71, 5, 63syl 20 . . . . . 6  |-  ( ph  ->  ( x  e.  (
<. V ,  E >. Neighbors  X
)  ->  x  e.  V ) )
84, 7syl5bi 217 . . . . 5  |-  ( ph  ->  ( x  e.  D  ->  x  e.  V ) )
98imp 429 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  V )
10 frgrancvvdeq.y . . . . 5  |-  ( ph  ->  Y  e.  V )
1110adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  V )
12 frgrancvvdeq.ny . . . . . 6  |-  N  =  ( <. V ,  E >. Neighbors  Y )
13 frgrancvvdeq.x . . . . . 6  |-  ( ph  ->  X  e.  V )
14 frgrancvvdeq.ne . . . . . 6  |-  ( ph  ->  X  =/=  Y )
15 frgrancvvdeq.xy . . . . . 6  |-  ( ph  ->  Y  e/  D )
16 frgrancvvdeq.a . . . . . 6  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
173, 12, 13, 10, 14, 15, 1, 16frgrancvvdeqlem1 30791 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
18 eldif 3449 . . . . . 6  |-  ( Y  e.  ( V  \  { x } )  <-> 
( Y  e.  V  /\  -.  Y  e.  {
x } ) )
19 ssnid 4017 . . . . . . . . 9  |-  x  e. 
{ x }
20 eleq1 2526 . . . . . . . . . 10  |-  ( Y  =  x  ->  ( Y  e.  { x } 
<->  x  e.  { x } ) )
2120eqcoms 2466 . . . . . . . . 9  |-  ( x  =  Y  ->  ( Y  e.  { x } 
<->  x  e.  { x } ) )
2219, 21mpbiri 233 . . . . . . . 8  |-  ( x  =  Y  ->  Y  e.  { x } )
2322necon3bi 2681 . . . . . . 7  |-  ( -.  Y  e.  { x }  ->  x  =/=  Y
)
2423adantl 466 . . . . . 6  |-  ( ( Y  e.  V  /\  -.  Y  e.  { x } )  ->  x  =/=  Y )
2518, 24sylbi 195 . . . . 5  |-  ( Y  e.  ( V  \  { x } )  ->  x  =/=  Y
)
2617, 25syl 16 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  x  =/=  Y )
279, 11, 263jca 1168 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
x  e.  V  /\  Y  e.  V  /\  x  =/=  Y ) )
28 frgraunss 30755 . . 3  |-  ( V FriendGrph  E  ->  ( ( x  e.  V  /\  Y  e.  V  /\  x  =/=  Y )  ->  E! y  e.  V  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )
292, 27, 28sylc 60 . 2  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  V  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E )
30 prex 4645 . . . . . . . . . . . 12  |-  { x ,  y }  e.  _V
31 prex 4645 . . . . . . . . . . . 12  |-  { y ,  Y }  e.  _V
3230, 31prss 4138 . . . . . . . . . . 11  |-  ( ( { x ,  y }  e.  ran  E  /\  { y ,  Y }  e.  ran  E )  <->  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E )
33 simpr 461 . . . . . . . . . . 11  |-  ( ( { x ,  y }  e.  ran  E  /\  { y ,  Y }  e.  ran  E )  ->  { y ,  Y }  e.  ran  E )
3432, 33sylbir 213 . . . . . . . . . 10  |-  ( { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E  ->  { y ,  Y }  e.  ran  E )
3534ad2antll 728 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  D )  /\  (
y  e.  V  /\  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )  ->  { y ,  Y }  e.  ran  E )
3612a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  N  =  ( <. V ,  E >. Neighbors  Y
) )
3736eleq2d 2524 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  e.  N  <->  y  e.  ( <. V ,  E >. Neighbors  Y ) ) )
38 nbgraeledg 23513 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( y  e.  ( <. V ,  E >. Neighbors  Y )  <->  { y ,  Y }  e.  ran  E ) )
391, 5, 383syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  e.  (
<. V ,  E >. Neighbors  Y
)  <->  { y ,  Y }  e.  ran  E ) )
4037, 39bitrd 253 . . . . . . . . . . 11  |-  ( ph  ->  ( y  e.  N  <->  { y ,  Y }  e.  ran  E ) )
4140adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  D )  ->  (
y  e.  N  <->  { y ,  Y }  e.  ran  E ) )
4241adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  D )  /\  (
y  e.  V  /\  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )  ->  ( y  e.  N  <->  { y ,  Y }  e.  ran  E ) )
4335, 42mpbird 232 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  D )  /\  (
y  e.  V  /\  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )  ->  y  e.  N
)
44 simpl 457 . . . . . . . . . 10  |-  ( ( { x ,  y }  e.  ran  E  /\  { y ,  Y }  e.  ran  E )  ->  { x ,  y }  e.  ran  E )
4532, 44sylbir 213 . . . . . . . . 9  |-  ( { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E  ->  { x ,  y }  e.  ran  E )
4645ad2antll 728 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  D )  /\  (
y  e.  V  /\  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )  ->  { x ,  y }  e.  ran  E )
4743, 46jca 532 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  (
y  e.  V  /\  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )  ->  ( y  e.  N  /\  { x ,  y }  e.  ran  E ) )
4847ex 434 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
( y  e.  V  /\  { { x ,  y } ,  {
y ,  Y } }  C_  ran  E )  ->  ( y  e.  N  /\  { x ,  y }  e.  ran  E ) ) )
4912eleq2i 2532 . . . . . . . . . . . . 13  |-  ( y  e.  N  <->  y  e.  ( <. V ,  E >. Neighbors  Y ) )
5049, 39syl5bb 257 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  e.  N  <->  { y ,  Y }  e.  ran  E ) )
5150biimpd 207 . . . . . . . . . . 11  |-  ( ph  ->  ( y  e.  N  ->  { y ,  Y }  e.  ran  E ) )
5251adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  D )  ->  (
y  e.  N  ->  { y ,  Y }  e.  ran  E ) )
5352impcom 430 . . . . . . . . 9  |-  ( ( y  e.  N  /\  ( ph  /\  x  e.  D ) )  ->  { y ,  Y }  e.  ran  E )
54 nbgraisvtx 23514 . . . . . . . . . . . . . . . 16  |-  ( V USGrph  E  ->  ( y  e.  ( <. V ,  E >. Neighbors  Y )  ->  y  e.  V ) )
551, 5, 543syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( y  e.  (
<. V ,  E >. Neighbors  Y
)  ->  y  e.  V ) )
5649, 55syl5bi 217 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( y  e.  N  ->  y  e.  V ) )
5756adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  D )  ->  (
y  e.  N  -> 
y  e.  V ) )
5857impcom 430 . . . . . . . . . . . 12  |-  ( ( y  e.  N  /\  ( ph  /\  x  e.  D ) )  -> 
y  e.  V )
5958ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( { y ,  Y }  e.  ran  E  /\  ( y  e.  N  /\  ( ph  /\  x  e.  D ) ) )  /\  {
x ,  y }  e.  ran  E )  ->  y  e.  V
)
60 simpl 457 . . . . . . . . . . . . 13  |-  ( ( { y ,  Y }  e.  ran  E  /\  ( y  e.  N  /\  ( ph  /\  x  e.  D ) ) )  ->  { y ,  Y }  e.  ran  E )
61 id 22 . . . . . . . . . . . . 13  |-  ( { x ,  y }  e.  ran  E  ->  { x ,  y }  e.  ran  E
)
6260, 61anim12ci 567 . . . . . . . . . . . 12  |-  ( ( ( { y ,  Y }  e.  ran  E  /\  ( y  e.  N  /\  ( ph  /\  x  e.  D ) ) )  /\  {
x ,  y }  e.  ran  E )  ->  ( { x ,  y }  e.  ran  E  /\  { y ,  Y }  e.  ran  E ) )
6362, 32sylib 196 . . . . . . . . . . 11  |-  ( ( ( { y ,  Y }  e.  ran  E  /\  ( y  e.  N  /\  ( ph  /\  x  e.  D ) ) )  /\  {
x ,  y }  e.  ran  E )  ->  { { x ,  y } ,  { y ,  Y } }  C_  ran  E
)
6459, 63jca 532 . . . . . . . . . 10  |-  ( ( ( { y ,  Y }  e.  ran  E  /\  ( y  e.  N  /\  ( ph  /\  x  e.  D ) ) )  /\  {
x ,  y }  e.  ran  E )  ->  ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E ) )
6564ex 434 . . . . . . . . 9  |-  ( ( { y ,  Y }  e.  ran  E  /\  ( y  e.  N  /\  ( ph  /\  x  e.  D ) ) )  ->  ( { x ,  y }  e.  ran  E  ->  ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E ) ) )
6653, 65mpancom 669 . . . . . . . 8  |-  ( ( y  e.  N  /\  ( ph  /\  x  e.  D ) )  -> 
( { x ,  y }  e.  ran  E  ->  ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E ) ) )
6766impancom 440 . . . . . . 7  |-  ( ( y  e.  N  /\  { x ,  y }  e.  ran  E )  ->  ( ( ph  /\  x  e.  D )  ->  ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E ) ) )
6867com12 31 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
( y  e.  N  /\  { x ,  y }  e.  ran  E
)  ->  ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E ) ) )
6948, 68impbid 191 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  (
( y  e.  V  /\  { { x ,  y } ,  {
y ,  Y } }  C_  ran  E )  <-> 
( y  e.  N  /\  { x ,  y }  e.  ran  E
) ) )
7069eubidv 2285 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  ( E! y ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E )  <->  E! y
( y  e.  N  /\  { x ,  y }  e.  ran  E
) ) )
7170biimpd 207 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( E! y ( y  e.  V  /\  { {
x ,  y } ,  { y ,  Y } }  C_  ran  E )  ->  E! y ( y  e.  N  /\  { x ,  y }  e.  ran  E ) ) )
72 df-reu 2806 . . 3  |-  ( E! y  e.  V  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E  <->  E! y ( y  e.  V  /\  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E ) )
73 df-reu 2806 . . 3  |-  ( E! y  e.  N  {
x ,  y }  e.  ran  E  <->  E! y
( y  e.  N  /\  { x ,  y }  e.  ran  E
) )
7471, 72, 733imtr4g 270 . 2  |-  ( (
ph  /\  x  e.  D )  ->  ( E! y  e.  V  { { x ,  y } ,  { y ,  Y } }  C_ 
ran  E  ->  E! y  e.  N  { x ,  y }  e.  ran  E ) )
7529, 74mpd 15 1  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  N  {
x ,  y }  e.  ran  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E!weu 2262    =/= wne 2648    e/ wnel 2649   E!wreu 2801    \ cdif 3436    C_ wss 3439   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403    |-> cmpt 4461   ran crn 4952   iota_crio 6163  (class class class)co 6203   USGrph cusg 23436   Neighbors cnbgra 23501   FriendGrph cfrgra 30748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-hash 12224  df-usgra 23438  df-nbgra 23504  df-frgra 30749
This theorem is referenced by:  frgrancvvdeqlem4  30794  frgrancvvdeqlem5  30795
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